Theorem clsndisj 7706

Description: Any open set containing a point that belongs to the closure of a subset intersects the subset. One direction of Theorem 6.5(a) of [Munkres] p. 95.

Hypothesis

Ref	Expression
clscld.1	|- X = U.J

Assertion

Ref	Expression
clsndisj	|- (((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) /\ (U e. J /\ P e. U)) -> (U i^i S) =/= (/))

Proof of Theorem clsndisj

Step	Hyp	Ref	Expression
1		eleq2 1535	. . . . 5 |- (x = U -> (P e. x <-> P e. U))
2		ineq1 2210	. . . . . 6 |- (x = U -> (x i^i S) = (U i^i S))
3	2	neeq1d 1594	. . . . 5 |- (x = U -> ((x i^i S) =/= (/) <-> (U i^i S) =/= (/)))
4	1, 3	imbi12d 626	. . . 4 |- (x = U -> ((P e. x -> (x i^i S) =/= (/)) <-> (P e. U -> (U i^i S) =/= (/))))
5	4	rcla4cv 1874	. . 3 |- (A.x e. J (P e. x -> (x i^i S) =/= (/)) -> (U e. J -> (P e. U -> (U i^i S) =/= (/))))
6	5	imp32 363	. 2 |- ((A.x e. J (P e. x -> (x i^i S) =/= (/)) /\ (U e. J /\ P e. U)) -> (U i^i S) =/= (/))
7		clscld.1	. . . . 5 |- X = U.J
8	7	elcls 7704	. . . 4 |- ((J e. Top /\ S (_ X /\ P e. X) -> (P e. ((cls` J)` S) <-> A.x e. J (P e. x -> (x i^i S) =/= (/))))
9	8	biimpa 416	. . 3 |- (((J e. Top /\ S (_ X /\ P e. X) /\ P e. ((cls` J)` S)) -> A.x e. J (P e. x -> (x i^i S) =/= (/)))
10		3simp1 788	. . . 4 |- ((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) -> J e. Top)
11		3simp2 789	. . . 4 |- ((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) -> S (_ X)
12	7	clsss3 7691	. . . . . 6 |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) (_ X)
13	12	sseld 2067	. . . . 5 |- ((J e. Top /\ S (_ X) -> (P e. ((cls` J)` S) -> P e. X))
14	13	3impia 830	. . . 4 |- ((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) -> P e. X)
15	10, 11, 14	3jca 819	. . 3 |- ((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) -> (J e. Top /\ S (_ X /\ P e. X))
16		3simp3 790	. . 3 |- ((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) -> P e. ((cls` J)` S))
17	9, 15, 16	sylanc 471	. 2 |- ((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) -> A.x e. J (P e. x -> (x i^i S) =/= (/)))
18	6, 17	sylan 448	1 |- (((J e. Top /\ S (_ X /\ P e. ((cls` J)` S)) /\ (U e. J /\ P e. U)) -> (U i^i S) =/= (/))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645   i^i cin 2046   (_ wss 2047  (/)c0 2280  U.cuni 2503  ` cfv 3182  Topctop 7588  clsccl 7662

This theorem is referenced by:  neindisj 7731

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-top 7592  df-cld 7663  df-ntr 7664  df-cls 7665

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